It is shown that in a plane with a radial density the four vertex theorem holds for the class of all simple closed curves if and only if the density is constant. On the other hand, for the class of simple closed curves that are invariant under a rotation about the origin, the four vertex theorem holds for every radial density.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm113-1-10,
author = {Doan The Hieu and Tran Le Nam},
title = {On the four vertex theorem in planes with radial density $e^{$\phi$(r)}$
},
journal = {Colloquium Mathematicae},
volume = {111},
year = {2008},
pages = {169-174},
zbl = {1152.53042},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm113-1-10}
}
Doan The Hieu; Tran Le Nam. On the four vertex theorem in planes with radial density $e^{φ(r)}$
. Colloquium Mathematicae, Tome 111 (2008) pp. 169-174. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm113-1-10/